The Stochastic Full Balance Sheet ModelBill Curry

British Actuarial Journal volume 24

Billcurry986@hotmail.com

13 December 2019Simplifying Actuarial Contractor Recruitment

Surplus projections

2

1in10 1yr

surplus

changes

Key questions in risk management

• What is the cost to our solvency position of a 20% equity fall?

• What fall in solvency is a 1in10 loss?

• What is the probability our capital coverage ratio goes below 120%?

• What is the probability we breach SCR?

• What event is most likely to cause a breach of SCR?

3

Risk appetite• Risk appetite buffers take into account the stability of the capital position

4

Types of group model

5

Regulatory

Capital

Economic

Capital

Full Balance

Sheet Model

Group model componentsRegulatory Capital – SII Standard Formula

6

Group Model

Model Structure

Risk Model Loss Function

Group model componentsRegulatory capital – SII Internal Model

7

Risk Model Loss Function

• Majority of firms use copula simulation

models

• Individual distributions specified for each

risk

• Losses to assets and liabilities normally

estimated through proxy functions

Model Structure• Model structure part

prescribed under SII

Group model componentsRegulatory capital – SII Standard Formula

8

Risk Model Loss Function

Model Structure• Model structure mostly

prescribed under SII

• No model actually specified. May be

thought of as multivariate normal

• Losses represented by linear loss

functions fitted to the 1in200 points

• No cross terms to represent interaction

between risks

Group model componentsEconomic capital

9

Risk Model Loss Function

Model Structure• Model structure not

prescribed, however

commonly similar to the

regulatory capital model

• May be similar to a firm’s regulatory model

• Other risks could be included

• Different calibrations could be used.

• Commonly similar to the regulatory model

Differences could be in:

• MA or VA

• Pension valuation

• Contract boundaries

• Etc.

Group model componentsStochastic full balance sheet model

10

Risk Model Loss Function

Model Structure• Structure may be close to

economic capital model

• Represents a firms best view of risks

• Likely to be aligned to economic capital

model

• Losses represent changes in the full SII

balance sheet rather than just assets and

liabilities

• Need to allow for realistic changes in

discount rates (VA, MA, IAS19)

Group model componentsStochastic full balance sheet model

11

-8

-3

2

-100%-50% 0% 50% 100%

Generate Sims Estimate Losses Analysis

Types of group model

12

Regulatory

Capital

Full Balance

Sheet Model

Strength of the Capital

Position

Stability of the Capital

Position

Model Features

Simulation generation

14

• Simulation generation may use standard copula modelling techniques

We should consider

• Should risk calibrations be Point In Time or Through The Cycle?

Proxy models

15

• The purpose of a proxy model is to enable fast estimation of balance sheet changes as a function of risk

movements.

Proxy models

16

• The purpose of a proxy model is to enable fast estimation of balance sheet changes as a function of risk

movements.

Roll forwards

17

• Roll forwards techniques are used to estimate how loss functions change

• Interest rate example,

Change in NAV = X – 20X2 Say we have a 1.6% interest rate increase

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

-2.0% -1.0% 0.0% 1.0% 2.0% 3.0%

Ch

an

ge i

n N

AV

Change in Interest Rates

Roll forwards

18

New Change in NAV = (X+1.6%) – 20(X+1.6%) 2

= 0.36X – 20X2 + 0.01088

The solvency II balance sheet

19

Assets

Transitional

Measures

BEL

SCR

Risk Margin

Surplus

Asset and liability modelling

20

• Asset and liability models typically the same as used in an Internal Model or Economic Capital model

Difficulties may arise over discount rates used for:

• Volatility Adjustment (VA) business

• Matching Adjustment (MA) business

• Pension liabilities

Example model

Annuity example - loss model

22

• 100,000 60 year old annuitants

• Annuity amount £1000 p.a.

• Expenses of £100, inflating at 1% p.a.

• Mortality as per an example mortality table

• Yield curve flat at 2%

• Risk free fixed interest cash-flows to broadly match the liability run off

Annuity example - risk model

23

• Normally distributed risks assumed for

– Longevity

– Expense

– Inflation

– Interest Rate PC1

– Interest Rate PC2

– Interest Rate PC3

– Credit

• Risks aggregated using a Gaussian copula with specified correlations

Longevity Inflation Expense PC1 PC2 PC3 Credit

Longevity 100% 0% 0% 0% 0% 0% 0%

Inflation 0% 100% 0% 50% 0% 0% -20%

Expense 0% 0% 100% 0% 0% 0% 0%

PC1 0% 50% 0% 100% 0% 0% -25%

PC2 0% 0% 0% 0% 100% 0% 0%

PC3 0% 0% 0% 0% 0% 100% 0%

Credit 0% -20% 0% -25% 0% 0% 100%

Annuity example - principal components analysis

24

• PCA is a dimension reduction technique commonly used to model yield curve changes

Term1

Term2

Term3

Term4

Term5

Term6

Term7

Term8

Term9

Term10

Term11

Term13

Term14

Term15

Term16

Term17

Term12 Term18

PC3

PC2

PC1

Annuity example – fitting approach

25

• Use Sobol sequence of 1023 fitting points

• Polynomial terms up to order 4 used to fit function to net assets

• Step regression applied to fit the proxy functions

• Out Of Sample (OOS) testing carried out using 100 random points from the risk model.

Proxy change in NAV =

f(L,I,E,PC1,PC2,PC3)

Annuity example – key exposures

26

-100

-50

0

50

100

-20% -10% 0% 10% 20%

Ch

an

ge i

n N

AV

(£m

)

Change in qx

Longevity

Annuity example – joint exposure PC1 vs longevity

27

Longevity

(change in qx)

0%

Threat direction

15%

-15%

-30%

SCR Proxy Modelling

SCR proxy models – SF

29

Generate proxy

functions for each run

Use proxy function to get

SF capitals

Fit SCR proxy function

OOS testing

Calibration

runs

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

-2.0% 0.0% 2.0%

Ch

an

ge i

n N

AV

Change in Interest Rates

Roll Forward

Process

Equity

Interest

SCR proxy models – Standard Formula

30

SCR Risk Model

Equity Risk

Model

Interest Rate

Risk Model

SCR proxy models – Standard Formula - longevity

31

SF stress

SF loss

• SF Longevity stress is a 20% fall in qx

SF Longevity Capital Estimate = -f(-20%,0,0,0,0,0)

SCR proxy models – Standard Formula - expense

32

• SF Expense stress is a 10% increase in expenses, together with a 1% increase in expense inflation

• Our example model uses a separate expense level risk and inflation risk

• Estimate SF expense capital using a combined expense and inflation event

SF Expense Capital Estimate = -f(0,1%,10%,0,0,0)

SCR proxy models – Standard Formula – interest rates

33

• SF Interest Rate up and down stresses are a function of the current yield curve

• We may estimate any change in yield curve as a linear combination of our principle components

Example, at yields of 2%, SF Yield down Capital Estimate = -f(0,0,0,-1.23,0.25,-1.16)

SCR proxy models – Standard Formula

34

SCR proxy models – Standard Formula – key exposures

35

-40

-20

-

20

40

-20% -10% 0% 10% 20%

Ch

an

ge in

NA

V

qx stress

Longevity

-40

-20

-

20

40

-50% -30% -10% 10% 30% 50%

Ch

an

ge in

NA

V

Expense stress

Expense

SCR proxy models – Internal Model

36

Generate proxy

functions for each run

Run SCR model for each run

Fit SCR proxy function

OOS testing

Calibration

runs

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

-2.0% 0.0% 2.0%

Ch

an

ge i

n N

AV

Change in Interest Rates

Roll Forward

Process

SCR proxy models – Internal Model

37

SCR proxy models – Internal Model key exposures

38

-60

-40

-20

0

20

40

60

-20% -10% 0% 10% 20%

NA

V c

ha

ng

e

qx stress

LongevitySF

IM

-80

-60

-40

-20

0

20

40

60

80

-50% -30% -10% 10% 30% 50%

NA

V c

ha

ng

e

Expense stress

Expense

Risk Margin Proxy Modelling

Risk Margin proxy models

40

Estimate changes in

RM for calibration set

Fit proxy function

OOS testing for fit

performance

Changes in

Discount rates

No VA/MA in Risk

Margin

Changes in Run-off

Risk Margin proxy models – run off example

41

• Annuity BEL run off under stress

Risk Margin proxy models – Standard Formula

42

Standard Formula example – key exposures

43

-140

-90

-40

10

60

110

-20% -10% 0% 10% 20%

NA

V C

ha

ng

e

qx stress

Longevity

-140

-90

-40

10

60

110

-50% -30% -10% 10% 30% 50%

NA

V C

ha

ng

e

Expense Stress

Expense

Discount Rates modelling

Liability modelling – Volatility Adjustment

45

• VA represents a flat addition to the discount curve for applicable long term liabilities

• Designed to protect insurers from the impact of volatility on the insurer’s solvency position

• Calculated as 65% of the spread between the interest rate of the assets in a reference portfolio and the risk

free rate, allowing for a fundamental spread

• Published monthly by EIOPA

• Permitted to change under SCR stress by some European supervisors, more recently in the UK

Liability modelling – Volatility Adjustment

46

Assets Liabilities

Credit stress event

SurplusSurplus

• How it works in practice

Liability modelling – Volatility Adjustment

47

Surplus Negative

Surplus

• How it works under non-dynamic VA SCR calculations

Liabilities

unchanged

Liability modelling – dynamic VA model

48

• VA ~ 65% x (Spread – Fundamental Spread) calculated by rating, maturity

Liability modelling – dynamic VA model

49

Liability modelling

50

Matching

Adjustment

IAS19 Discount

Rates

• We need realistic models to take into

account the movement of these under stress

Annuity example - dynamic VA, SF model

51

Include credit risk as an additional normally distributed risk in the model.

Annuity example - dynamic VA, SF model

52

Repeat curve fit process for net assets and SCR, RM unchanged

-60

-40

-20

-

20

40

60

80

-60 -40 -20 - 20 40 60 80Pro

xy (

£m

)

Actual (£m)

SCR Proxy Fit

Annuity example - dynamic VA, SF model

53

Low materiality changes in other risk exposures

Using the Example Model

More assumptions

55

• Starting surplus = £250m

• Risk Appetite thresholds (based on a one year time frame):

– Plan to be able to withstand a 1in30 shock

– We take urgent action if our surplus is unable to withstand a 1in10 shock

Risk appetite

56

-600 -500 -400 -300 -200 -100 0 100 200 300 400 500

Den

sit

y

Change in NAV (£m)

1in10

1in30

Ruin probabilities

57

-600 -500 -400 -300 -200 -100 0 100 200 300 400 500

Den

sit

y

Change in NAV (£m)

Ruin

Current

Surplus

Ruin Prob = 5.9%

Euler allocation by risk – ruin event of £250m loss

58

• Euler Allocation of 250m loss

Risk A allocation = −𝑬 𝑿𝑨 𝑿𝒕𝒐𝒕𝒂𝒍 = −𝟐𝟓𝟎𝒎}

Interest inflation credit longevity expense cross

4 72 262 -23 -25 -39

-10 65 -13 74 117 18

28 54 263 -47 -34 -13

16 136 100 5 13 -20

-16 255 76 -78 53 -40

-12 -64 223 41 71 -9

Ruin Events

Euler example

60

• Risks A and B, multivariate standard normal

• Correlation -99.9%

𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝑵𝑨𝑽 = 𝟐 − ( 𝒆𝑨+ 𝒆𝑩)

1in200 capital = 14.8m

Euler allocations:

Risk A = 7.4m

Risk B = 7.4m

Ruin events – Most Likely Ruin Event

61

0

0.2

0.4

0.6

0.8

1

1.2D

en

sit

y

Risk movement

MLRE

Ruin region

Ruin events – Most Likely Ruin Event

62

Ruin region

MLRE

Ruin events – Euler example

63

• Risk distribution is multivariate normal (-99.9% correlation)

• Density function of A,B is well known f(A,B)

• Can solve for the maximum of f(A,B) subject to constraint 𝟐 − ( 𝒆𝑨+ 𝒆𝑩) = -14.8

Max at (A,B) = (-2.8,2.8) and (2.8,-2.8)

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

-4 -3 -2 -1 0 1 2 3 4

Ch

an

ge i

n N

AV

Risk X

Ruin events – Euler example

64

A= -B

Ruin

region

Most likely

points

A

B

Ruin events – annuity model

65

• Find MLRE by maximising probability density subject to change in NAV < -£250m

• Risk distribution ~ Multivariate normal so density is well defined

• Change in NAV estimated using proxy functions.

Risk Movement Percentile 1 in X

PC1 0.437 0.67 3.0

PC2 - 0.209 0.42 2.4

PC3 - 0.128 0.45 2.2

Inflation 0.6% 0.78 4.6

Credit (spreads) - 1.8% 0.12 8.1

Longevity (qx) - 3.8% 0.31 3.2

Expense (level) 14.1% 0.77 3.0

Ruin events – annuity model

66

0

20

40

60

80

100

120

140

Co

st

(£m

)

Loss by Risk

Ruin events – Kernel Density Estimation

67

• We can use Kernel Density Estimation (KDE) to estimate the density function of the joint risk distribution

from the simulations

Ruin events – annuity model

68

Ruin cause

69

No. Event %

1 Credit 56

2 Credit / Inflation 11

3 Inflation 8

4 Expense 8

5 Credit / Expense 6

Others 11

Ruin cause

70

No. Event Credit (change

in spreads)

Inflation (change

in RPI)

Expense

(change in level)

1 Credit -3.5%

2 Credit / Inflation -2.6% 0.9%

3 Inflation 2.8%

4 Expense 68.7%

5 Credit / Expense -2.5% 19.6%

Ruin events summary

71

Euler is about capital allocation not events

MLREs give insight into actual events

KDE can be used to get the density from the simulations

We can plan what we would do under the ruin events

Uses of the model

The roll forward cycle

73

Recalibrate

Roll Forwards

Assess Roll forwards

performance

Roll Forwards for:

• Run off

• New Business

• Economics

• Basis Changes

• Model changes

• Risk calibration changes

Projections

74

• Is our balance sheet getting more or less stable over time?

• Is our ruin probability getting better or worse?

Projections

75

Proxy change in NAV = f(L,I,E,PC1,PC2,PC3)

• Proxy functions normally express changes in NAV as a function of risk movements

• For projections (e.g. of risk appetite), we need to calibrate as a function of risks

and time

Proxy change in NAV = f(L,I,E,PC1,PC2,PC3,Time)

• We can use run off drivers by risk and product to scale the proxy functions over

time

Projections - example

76

Proxy change in NAV = aX2 + bX + cY + d XY

• For Risks X and Y

What isn’t in your model?

77

Regulatory risk?

Changes to business

plan?

Liquidity risk?

Long term risks?

Regime changes?

Types of group model

78

Regulatory

Capital

Economic

Capital

Full Balance

Sheet Model

79

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The information and expressions of opinion contained in this presentation are not intended to be a comprehensive study, nor to provide actuarial advice or

advice of any nature and should not be treated as a substitute for specific advice concerning individual situations. On no account may any part of this

presentation be reproduced without the written permission of the IFoA.

Questions Comments